Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory

Multiscale kinetic inviscid flux extracted from a gas-kinetic scheme for simulating incompressible and compressible flows

A kinetic inviscid flux (KIF) is proposed for simulating incompressible and compressible flows. It is constructed based on the direct modeling of multiscale flow behaviors, which is used in the gas-kinetic scheme (GKS), the unified gas-kinetic scheme (UGKS), the discrete unified gas-kinetic scheme (DUGKS), etc. In KIF, the discontinuities (such as the shock wave) that cannot be well resolved by mesh cells are mainly solved by the kinetic flux vector splitting (KFVS) method representing the free transport mechanism (or microscale mechanism), while in other flow regions that are smooth, the flow behavior is solved mainly by the central-scheme-like totally thermalized transport (TTT). The weights of KFVS and TTT in KIF is automatically determined by those in the theory of direct modeling. Two ways of choosing the weights in KIF are proposed, which are actually the weights adopted in the UGKS and the DUGKS, respectively. By using the test cases of the Sod shock tube, the rarefaction wave, the boundary layer of a flat plate, the cavity flow, hypersonic flow over a circular cylinder, the shock and turbulent boundary iteration, and transonic flow over a three-dimensional M6 wing, the validity and accuracy of the present method are examined. The KIF does not suffer from the carbuncle phenomenon, and does not introduce extra numerical viscosity in smooth regions. Especially in the case of hypersonic cylinder, it gives quite sharp and clear density and temperature contours. The KIF can be viewed as an inviscid-viscous splitting version of the GKS. By doing this splitting, it is easy to be used in traditional computational fluid dynamics frameworks. It can also be classified as a type in the numerical schemes based on the kinetic theory that are represented by the works of Sun et al. [Adv. Appl. Math. Mech. 8, 703 (2016)10.4208/aamm.2015.m1071] and Ohwada et al. [J. Comput. Phys. 362, 131 (2018)JCTPAH0021-999110.1016/j.jcp.2018.02.019], except the weights are determined by the weights of direct modeling.